1 1 y = y() y, y,..., y (n) : n y F (, y, y,..., y (n) ) = 0 n F (, y, y ) = 0 1 y() 1.1 1 y y = G(, y) 1.1.1 1 y, y y + p()y = q() 1 p() q() (q() = 0) y + p()y = 0 y y + py = 0 y y = p (log y) = p log y = p( )d = P () + C y = C e p( )d p() P () p() C(= e C) C y(0) = y 0 p( )d
y 2y = 0 (log y) = 2 log y = 2 d = 2 + C y = Ce 2 y(0) = y 0 C = y 0 t N(t) dn dt = λn λ > 0 N(0) = N 0 N(t) = N 0 e λt N(t 0 ) = N 0 /2 t 0 N 0 2 = N 0e λt0 e λt0 = 1 2 1.1 137 Cs 131 I 30 [year] 8.04 [day] (log 2 0.6931 2011 3 137 Cs ep( t) 1.00 0.98 0.96 0.94 0.92 0.00 0.02 0.04 0.06 0.08 0.10
y + py = q (log y) = p d ( ) = d y + py = ( ) µ() µp = µ µ y + µp y = µ q ( ) = µy + µ y = (µy), (µy) = µq µ() µ = pµ µ() µy = µqd + C y = 1 ( µqd + C) µ µ = ep p( )d ( = e ) p( )d e p( )d q( )d + C } {{ } =C() y = C()e p( )d C C() y = C()e p( )d C() y = C e p( )d py y + py = C e p( )d = q C = e p( )d q C() = e p( )d q( )d + C y + y = 1 y + y = 0 y = Ce C C() y = y + C e, C e = 1 C = e C() = e + C y = (e + C) e = 1 + Ce 1.2 (1) dy d + y cos = 0 (2) dy d + y = e (3) dy d + (4) dy d + 2 y = 1 2 1 + 2 y = 1 1 + 2
2 1.1.2 1 y = F (, y) F (, y) y = f(y) dy f(y) = F (, y) y f(y) g() y = f(y)g() dy f(y) = d g()d y = 2y 2 dy y 2 = 2 d 1 y = 2 + C y = 1 2 + C p(t) dp dt (1) dp dt = ap, a ( () a > 0 p (2) dp dt p2 logistic dp dt = ap bp2 a, b 2.1 (1) dy d = 32 y (2) (1 + 2 ) dy d = 1 + tan α + tan β y2, ( tan(α + β) = 1 tan α tan β 2.2 p(t) dp dt = ap bp2 a, b (t = 0) p 0 a = 0.029, b = 2.695 10 12
1.1.3 1 y = f ( y ) v := y/ v = y y 2 y = 2 v + y = v + v v + v = f(v) v = f(v) v v dv d f(v) v = y = + 2y 2 + y 1 + 2v = 2 + v v = 1 + v2 2 + v v + 2 1 + v 2 dv = log + C 2 tan 1 v + 1 2 log(1 + v2 ) = log + C ( 4 tan 1 y ) + log( 2 + y 2 ) = C y y 1 1.1.4 Bernoulli 1 y + p()y = q()y n Bernoulli n = 0 n = 1 n 0, 1 n Bernoulli y n y y n + p 1 y n 1 = q u = 1/yn 1 u = (1 n) y y n y y n + p 1 u = q yn 1 1 n + pu = q u + (1 n)pu = (1 n)q u 2y + y = 2 2 ( + 1)y 3 y + 1 y = ( + 1)y3 2 n = 3 Bernoulli u = 1/y 2 u 1 u = 2( + 1) p() = 1/ q = 2( + 1) u() = C()e pd pd = d C() = = log e pd = e log = 1 ( 2) ( + 1)d = 2 ( + 1)d = 2 2 u() = ( 2 2 + C) = 3 2 2 + C 1 u = y2 = 1 3 + 2 2 C 2.3 (1) dy d = 2 + y 2 2y (2) 3 dy d y y2 = 0
3 1.1.5 1 y P (, y)d + Q(, y)dy = 0 ( P (, y) + Q(, y)y = 0) ϕ(, y) dϕ(, y) = ϕ ϕ d + y dy = 0 ϕ(, y) = C C ϕ y() d dy t ((t), y(t)) dϕ dϕ = ( ) ϕ ϕ d + y dy = ( ϕ d dt + ϕ ) dy dt y dt }{{} =0 y dϕ = 0 0 ϕ = C 2d + 2ydy = 0 + yy = 0 yy = 2d + 2ydy = 2 + y 2 = C 2 ϕ(, y) = 2 + y 2 y = C cos t, y = C sin t P d + Qdy = 0 2 ϕ(, y) P (, y) = ϕ ϕ (, y), Q(, y) = (, y) y P y = 2 ϕ y = 2 ϕ y = Q P y = Q P =, Q = y P y = Q = 0 a > 0 P = 3 2 3ay, Q = 3y 2 3a (3 2 3ay)d + (3y 2 3a)dy = 0, (, y) = (0, 0) P y = 3a, Q = 3a
P = ϕ ϕ = (3 2 3ay)d = 3 3ay + φ(y) ϕ y = 3a + φ (y) = 3y 2 3a φ(y) = y 3 ϕ(, y) = 3 3ay + y 3 = C C = 0 P d + Qdy = 0 µ(, y) µ(p d + Qdy) = 0 µ y (µp ) = ( P (µq) µ y Q ) = µ Q µ y P µ(p d + Qdy) = 0 µ µ Q(, y) = 1 P y µ = dµ d µ µ = P y P y P (, y) P = p()y + q() y (p()y + q())d + dy = 0 dy = (py + q) d 1 1 1 3.1 y(e) = 1 ( 1 1 ) d + 1 y y 2 dy = 0 3.2 1.2 (1) dy d + y cos = 0 (2) dy d + y = e (3) dy d + (4) dy d + 2 y = 1 2 1 + 2 y = 1 1 + 2 1.2 2 2 F (, y, y, y ) = 0 y (F (, y, y ) = 0) y = p p (F (y, y, y ) = 0) y = p y = p = p dy y d = p y p F ( y, p, p ) y p = 0 y p(y)
1.2.1 2 y = G(, y, y ) y + p()y + q()y = r() r() = 0 y + p()y + q()y = 0 p() q() y( 0 ) = y 0, y ( 0 ) = y 0 y 1 y 2 y 1 y 2 y 1y 2 0 C 1 C 2 y = C 1 y 1 + C 2 y 2 = 0 y( 0 ) = y 0 = C 1 y 1 ( 0 ) + C 2 y 2 ( 0 ) y ( 0 ) = y 0 = C 1 y 1( 0 ) + C 2 y 2( 0 ) 2 ( ) ( ) y1 ( 0 ) y 2 ( 0 ) C1 y 1( 0 ) y 2( = 0 ) C 2 y 1 y 2 y 1y 2 0 2 2 ( C1 C 2 ) = 1 y 1 ( 0 )y 2 ( 0) y 1 ( 0)y 2 ( 0 ) ( y0 y 0 ) ( y 2 ( 0 ) y 2 ( 0 ) y 1( 0 ) y 1 ( 0 ) ) ( ) y0 C 1, C 2 y 1 y 2 y 1y 2 0 2 y 0, y 0 2 C 1, C 2 y 0 y = C 1 y 1 + C 2 y 2 2
4 W [y 1, y 2 ] := y 1 y 2 y 1 y 2 = y 1y 2 y 1y 2 0 W [y 1, y 2 ] Wronskian y 2 y 1 Wronskian 0 Wronskian 0 y 1 y 2 Wronskian 1 W [y 1, y 2 ] = (y 1 y 2 y 1y 2 ) = y 1 y 2 y 1 y 2 = y 1 ( py 2 qy 2 ) ( py 1 qy 1 )y 2 = p()(y 1 y 2 y 1y 2 ) = p()w [y 1, y 2 ] y 1, y 2 1 W [y 1, y 2 ] = Ce p( )d C = 0 0 y + y = 0 2 y 1 = sin y 2 = cos W [sin, cos ] = y 1 y 2 y 1 y 2 = sin cos cos sin = 1 0 y 1 y 2 y + y = 0 y = C 1 sin + C 2 cos 4.1 2 2 y + 3y y = 0 (1) y 1 = y 2 = 1/ > 0 (2) y 1 y 2 (3) y(1) = 2, y (1) = 1 4.2 y 1, y 2 αy 1 + βy 2 γy 1 + δy 2 α,..., δ y + p()y + q()y = 0 y 1 y 1 y 2 = y 1 v v() y 2 + py 2 + qy 2 = (y 1 v + 2y 1v + y 1 v ) + p(y 1v + y 1 v ) + q(y 1 v) = (2y 1 + y 1 p)v + y 1 v = 0 u := v 1 y 1 u + (2y 1 + y 1 p)u = 0 y 1 ( ) 2y u = 1 + p u log u = 2 log y 1 pd + C y 1 u = C y1 2 e pd v = ud v y 2 = y 1 ud Wronskian W [y 1, y 2 ] = y 1 y 2 y 1y 2 = e pd Wronskian 0 y 1 y 2
2 y 6y = 0 y 1 = 3 y 2 = 2 y 1 y 2 p() = 0 u = C/y 2 1 u = 6 y 2 = y 1 d = 2 6 4.3 2 y (3 + 1)y + (2 + 1)y = 0 (1) y = e (2) (y = ( 1)e 2 ) y + py + qy = r 1 y + py + qy = 0 y 1, y 2 C 1, C 2 y() = C 1 y 1 () + C 2 y 2 () y y = C 1 y 1 + 2C 1y 1 + C 1 y 1 + C 2 y 2 + 2C 2y 2 + C 2 y 2 py = p(c 1 y 1 + C 1y 1 ) + p(c 2 y 2 + C 2y 2 ) qy = q C 1 y 1 + q C 2 y 2 y 1, y 2 2C 1y 1 + C 1 y 1 + 2C 2y 2 + C 2 y 2 + p(c 1y 1 + C 2y 2 ) = r C 1, C 2 C 1y 1 + C 2y 2 = 0 C 1 y 1 + C 2 y 2 + C 1y 1 + C 2y 2 = 0 y { C 1 y 1 + C 2y 2 = 0 C 1y 1 + C 2y 2 = r C 1y 1 + C 2y 2 = r ( ) ( ) y1 y 2 C 1 y 1 y 2 C 2 = C 1, C 2 2 2 y 1 y 2 Wronskian W [y 1, y 2 ] ( ) 0 r Cramer C 1 1 = W [y 1, y 2 ] 0 y 2 r y 2 = ry 2 W [y 1, y 2 ], 1 C 2 = W [y 1, y 2 ] y 1 0 y 1 r = ry 1 W [y 1, y 2 ] C 1 () = ry 2 ry 1 W [y 1, y 2 ] d, C 2 () = = W [y 1, y 2 ] d y + y = cos C 1 sin + C 2 cos W [sin, cos ] = 1 C 1 = cos2 C 1 = 1 1 2 C 2 cos sin = 1 C 2 = ( + 12 sin 2 ) + C 1 (1 + cos 2 )d = 1 2 cos sin d = 1 2 cos2 + C 2 C 1, C 2 2 C 1, C 2 y() = C 1 sin + C 2 cos + 1 ( + 12 ) 2 sin 2 sin + 1 2 cos3 = C 1 sin + C 2 cos + 1 2 sin
5 C 1, C 2 r() y + y + y = 2 y = a 0 + a 1 + a 2 2 2a 2 + (a 1 + 2a 2 ) + (a 0 + a 1 + a 2 2 ) = 2 (2a 2 + a 1 + a 0 ) + (2a 2 + a 1 ) + (a 2 1) 2 = 0 0 a 2 = 1, a 1 = 2, a 0 = 0 y = 2 + 2 y 2y + 4y = e 2 y = ae 2 (4a 4a + 4a)e 2 = e 2 4a = 1 a = 1 4 y = e 2 /4 y + 4y = sin 3 y = a sin 3 ( 9a + 4a) sin 3 = sin 3 a = 1 5 y = 1 5 sin 3 y + y = cos 1.2.2 2 y 2 y + 2ay + by = r a, b r y + 2ay + by = 0 y = m 3 y, y, y y y = e λ y = λe λ, y = λ 2 e λ y + 2ay + by = (λ 2 + 2aλ + b)e λ = 0 0 λ λ 2 + 2aλ + b = 0 y = e λ λ 2 λ ± := a ± a 2 b 2 a 2 b 0 y 1 = e λ +, y 2 = e λ Wronskian λ + λ W [e λ +, e λ ] = (λ λ + )e (λ ++λ ) 0 y 1, y 2 y() = C 1 e λ + + C 2 e λ
y y 6y = 0 λ 2 λ 6 = 0 (λ + 2)(λ 3) = 0 λ = 3, 2 y = C 1 e 3 + C 2 e 2 y + 2y + 4y = 0 λ 2 + 2λ + 4 = 0 λ ± = 2 ± 3i y 1 = e ( 2+ 3i) = e 2 (cos 3 + i sin 3) y 1 = e ( 2 3i) = e 2 (cos 3 i sin 3) C 1, C 2 y = (C 1 cos 3 + C 2 sin 3)e 2 a 2 b = 0 y + 2ay + a 2 y = 0 λ 2 + 2aλ + a 2 = 0 (λ + a) 2 = 0 λ = a y 1 = e a y 2 = y 1 v u = v u ( ) 2y u = 1 + p u y 1 p() = 2a 2y 1/y 1 = 2a 0 u = 0 v = 0 v 1 v = C v y 2 = y 1 = e a y = C 1 e a + C 2 e a 5.1 (1) 6y 7y + y = 0 (2) y 3y + y = 0 (3) y + y + y = 0 (4) y 6y + 9y = 0 5.2 (1) y + 3y = 3 1 (2) y + 2y 15y = cos 3 (3) y 2y 3y = e 3 (y = C 1 e + C 2 e 3 + e 3 /4)
6 Euler y + py + qy = 0 p() 1/ q() 1/ 2 2 y + α y + β 2 y = 0 2 y + αy + βy = 0 Euler Euler = e t t t = e t t = log dt d = 1 d d = dt d d dt = 1 d dt d 2 d 2 = d ( ) 1 d = 1 d d dt 2 dt + 1 1 d 2 dt 2 = 1 ( d 2 2 dt 2 d ) dt 2 y + αy + βy = 0 d2 y + (α 1)dy dt2 dt + βy = 0 (α 1) 2 4β 0 λ 2 + (α 1)λ + β = 0 λ ± t y(t) = C 1 e λ +t + C 2 e λ t 2 y + αy + βy = 0 y() = C 1 λ + + C 2 λ (α 1) 2 4β = 0 λ = α + 1 y(t) = C 1 e λt + C 2 te λt y() = C 1 λ + C 2 λ log Euler 2 y + αy + βy = 0 1.2.3 2 Euler p() = α/ q() = β/ 2 2 0 p() = p 1 q 2 + p 0 + p 1 ( 0 ) +, q() = 0 ( 0 ) 2 + q 1 + q 0 + q 1 ( 0 ) + 0 ( 0 )p() ( 0 ) 2 q() = 0 Taylor = 0 p() q() = 0 Taylor = 0
y + 2y + 2y = 0 = 0 p() = 2, q() = 2 = 0 y = a 0 + a 1 + a 2 2 + = n=1 a n n y = a 1 + 2a 2 + 3a 3 2 + = na n n 1, y = 2a 2 + 6a 3 + = n(n 1)a n n 2 y + 2y + 2y = 2a 2 + 2a 0 + n=0 n=2 (2a n + 2na n + (n + 2)(n + 1)a n+2 ) n n=1 0 n 0 a 2 + a 0 = 0, (2n + 2)a n + (n + 2)(n + 1)a n+2 = 0 a 2 = a 0, a n+2 = 2 n + 2 a n 2 n n = 2k a 2 = a 0, a 2k+2 = 1 k + 1 a 2k a 2k = ( 1)k a 0 a 0 = 1 y 0 k! y 0 = ( 1) k 2k = e 2 k! k=0 n = 2k + 1 a 2k+1 = ( 1)k 2 k a 2k+3 = 2 2k + 3 a 2k+1 (2k + 1)!! a 0 a 1 = 1 y 1 y 1 = k=0 ( 1) k 2 k (2k + 1)!! 2k+1 y 1 C 0, C 1 y = C 0 y 0 + C 1 y 1 y 0, y 1 Bessel 2 y + y + ( 2 ν 2 )y = 0 Bessel 2 p() = 1/ q() = 1 (ν/) 2 = 0 ν = 1/2 Taylor y = λ n=0 a n n λ y = λ λ 1 n=0 y = λ(λ 1) λ 2 a n n + λ n=0 n=1 na n n 1, a n n + 2λ λ 1 n=1 na n n 1 + λ n=2 n(n 1)a n n 2
2 y = λ(λ 1) λ y = λ λ ( 2 1 ) y = λ 4 n=0 n=2 n=0 a n n + λ a n n + 2λ λ a n 2 n 1 4 λ n=1 na n n, n=0 n=1 a n n na n n + λ n=2 n(n 1)a n n, 2 y + y + ( 2 1 4 )y = 0 n 0 ( λ 2 1 ) a 0 = 0, 4 ( λ 2 + 2λ + 3 ) a 1 = 0 4 { a n 2 + (λ + n) 2 1 } a n = 0 (n 2) 4 a 0 λ λ λ = ±1/2 6.1 ( 2 y + y + ( 2 1 4 )y = 0 (1) λ = 1/2 a 0 = 1 a 2n+1 = 0, a 2n = ( 1)n (2n + 1)! y 1 = 1 sin (2) λ = 1/2 a 0 = 1 a 1 = 0 y 2 = 1 cos 6.2 (1 2 )y 2y + α(α + 1)y = 0 Legendre I 7.4 (1) = ±1 (2) α = n Legendre n (3) α = n P n (1) = 1 Legendre P 0 (), P 1 (), P 2 (), P 3 ()